The plus symbol as an abbreviation for the Latin *et*, though appearing with the
downward stroke not quite vertical, was found in a manuscript dated
1417 (Cajori).

The + and - symbols first appeared in print in *Mercantile
Arithmetic* or *Behende und hüpsche Rechenung auff allen
Kauffmanschafft*, by Johannes Widmann (born c. 1460), published in
Leipzig in 1489. However, they referred not to addition or
subtraction or to positive or negative numbers, but to surpluses and
deficits in business problems (Cajori vol. 1, page 128).

**Here
is an image of the first use in print of the + and - signs,** from
Widman's *Behennde vnd hüpsche Rechnung.* This image is
taken from the Augsburg edition of 1526.

Widman wrote, "Was - ist / das ist minus ... vnd das + das ist mer." He also wrote, "4 centner + 5 pfund" and "5 centner — 17 pfund," thus showing the excess or deficiency in the weight of boxes or bales (Smith vol. 2, page 399).

Smith (vol. 2, page 398) explains the origin of the + sign by connecting it to the Latin word for "and":

In a manuscript of 1456, written in Germany, the wordetis used for addition and is generally written so that it closely resembles the symbol +. Theetis also found in many other manuscripts, as in "5 et 7" for 5 + 7, written in the same contracted form, as when we write the ligature & rapidly. There seems, therefore, little doubt that this sign is merely a ligature foret.

Cajori says, "There is clear evidence that, as a lecturer at the University of Leipzig, Widmann had studied manuscripts in the Dresden library in which + and - signify operations, some of these having been written as early as 1486." Johnson (page 144) says a series of notes from 1481, annotated by Widmann, contain the + and - symbols, and he asks whether Widman could have copied these symbols from some unknown professor at the University of Leipzig. Johnson also says that a student's notes from one of Widmann's 1486 lectures show the + and - signs.

Giel Vander Hoecke used + and - as symbols of operation in *Een
sonderlinghe boeck in dye edel conste Arithmetica,* published at
Antwerp in 1514 (Smith 1958, page 341). Burton (page 335) says Vander
Hoecke was the first person to use + and - in writing algebraic
expressions, but Smith (page 341) says he followed Grammateus.

Henricus Grammateus (also known as Henricus Scriptor and Heinrich
Schreyber or Schreiber) published an arithmetic and algebra, entitled
*Ayn new Kunstlich Buech,* printed in 1518, in which he used +
and - in a technical sense for addition and subtraction (Cajori vol.
1, page 131).

The plus and minus symbols only came into general use in England after
they were used by Robert Recorde in in 1557 in *The Whetstone of
Witte.* Recorde wrote, "There be other 2 signes in often use of
which the first is made thus + and betokeneth more: the other is thus
made - and betokeneth lesse."

The plus and minus symbols were in use before they appeared in print. For example, they were painted on barrels to indicate whether or not the barrels were full. Some have attempted to trace the minus symbol as far back as Heron and Diophantus.

X actually appears earlier, in
1618 in an anonymous appendix to Edward Wright's translation of John
Napier's *Descriptio* (Cajori vol. 1, page 197). However, this
appendix is believed to have been written by Oughtred.

**The dot (·)** was advocated by Gottfried
Wilhelm Leibniz (1646-1716). According to Cajori (vol. 1, page 267):

The dot was introduced as a symbol for multiplication by G. W. Leibniz. On July 29, 1698, he wrote in a letter to John Bernoulli: "I do not like X as a symbol for multiplication, as it is easily confounded withCajori shows the symbol as a raised dot. However, according to Margherita Barile, consulting Gerhardt's edition of Leibniz'sx; ... often I simply relate two quantities by an interposed dot and indicate multiplication byZC·LM.Hence, in designating ratio I use not one point but two points, which I use at the same time for division."

The dot was used earlier by Thomas Harriot (1560-1621) in
*Analyticae Praxis ad Aequationes Algebraicas Resolvendas,* which
was published posthumously in 1631, and by Thomas Gibson in 1655 in
*Syntaxis mathematica.* However Cajori says, "it is doubtful
whether Harriot or Gibson meant these dots for multiplication. They
are introduced without explanation. It is much more probable that
these dots, which were placed after numerical coefficients, are
survivals of the dots habitually used in old manuscripts and in early
printed books to separate or mark off numbers appearing in the running
text" (Cajori vol. 1, page 268).

However, Scott (page 128) writes that Harriot was "in the habit of using the dot to denote multiplication." And Eves (page 231) writes, "Although Harriot on occasion used the dot for multiplication, this symbol was not prominently used until Leibniz adopted it."

**The asterisk (*)** was used by Johann Rahn (1622-1676)
in 1659 in *Teutsche Algebra* (Cajori vol. 1, page 211).

**By juxtaposition.** In a manuscript found buried in
the earth near the village of Bakhshali, India, and dating to the
eighth, ninth, or tenth century, multiplication is normally indicated
by placing numbers side-by-side (Cajori vol. 1, page 78).

Multiplication by juxtaposition is also indicated in "some fifteenth-century manuscripts" (Cajori vol. 1, page 250). Juxtaposition was used by al-Qalasadi in the fifteenth century (Cajori vol. 1, page 230).

According to Lucas, Michael Stifel (1487 or 1486 - 1567) first showed
multiplication by juxtaposition in 1544 in *Arithmetica
integra.*

In 1553, Michael Stifel brought out a revised edition of Rudolff's
*Coss,* in which he showed multiplication by juxtaposition and
repeating a letter to designate powers (Cajori vol. 1, pages
145-147).

**The colon (:)** was used in 1633 in a text entitled
*Johnson Arithmetik; In two Bookes* (2nd ed.: London, 1633).
However Johnson only used the symbol to indicate fractions (for
example three-fourths was written 3:4); he did not use the symbol for
division "dissociated from the idea of a fraction" (Cajori vol. 1,
page 276).

Gottfried Wilhelm Leibniz (1646-1716) used : for both ratio and
division in 1684 in the *Acta eruditorum* (Cajori vol. 1, page
295).

**The obelus (÷)** was first used as a division symbol by
Johann Rahn (or Rhonius) (1622-1676) in 1659 in *Teutsche
Algebra* (Cajori vol. 2, page 211).

Here is the page in which the division symbol first appears in print, as reproduced in Cajori.

Rahn's book was translated into English and published, with additions by John Pell, in London in 1668, with the division symbol retained. According to some recent sources, John Pell was a major influence on Rahn and he may in fact be responsible for the invention of the symbol. However, according to Cajori there is no evidence to support this claim. The division symbol was used by many writers before Rahn as a minus sign.

**Recent symbolism.** In nineteenth century U. S. textbooks, long
division is typically shown with the divisor, dividend, and quotient
on the same line, separated by parentheses, as 36)116(3. See the figure below.

This same 1772 book shows a vinculum above the dividend when dividing decimals:

In 1882 in *Complete Graded Arithmetic* by James B. Thomson, the
36)116(3 notation is used for long division. However, in examples
for short division, a vinculum is placed under the dividend and the
vinculum is almost attached to the bottom of the close parenthesis.
The quotient is written under the vinculum, as shown below.

In 1888 in the teacher's edition of *The Elements of Algebra* by G. A.
Wentworth the vinculum is almost attached to the top of the close parenthesis
and the quotient is written above the vinculum, as shown below.

The symbolism shown above may also appear in the earlier 1882 edition of Wentworth, which has not been seen. I welcome earlier uses of this symbol that may be found by readers of this page.

In 1901, the second edition of *Robinson's Complete Arithmetic*
by Daniel W. Fish uses the same notations for short and long division
as Thomson (1882) above, except that the vinculum under the dividend
is actually attached to the close parenthesis. This notation may
appar in the earlier 1873 edition, which has not been seen.

David E. Smith writes, "It is impossible to fix an exact date for the origin of our present arrangement of figures in long division, partly because it developed gradually" (Smith vol. 2).

The symbol is not mentioned by Cajori. The symbol does not have a name.

Nicolas Chuquet (1445?-1500?) used raised numbers in *Le Triparty en
la Science des Nombres* in 1484. However, in Chuquet's notation,
12^{3} actually meant 12*x*^{3} (Cajori vol. 1,
page 102).

In 1634, Pierre Hérigone (or Herigonus) (1580-1643) wrote a, a2,
a3, etc., in *Cursus mathematicus,* which was published in
several volumes from 1634 to 1637; the numerals were not raised,
however (Cajori vol. 1, page 202, and Ball).

In 1636 James Hume used Roman numerals as exponents in
*L'Algèbre de Viète d'vne methode novelle, claire, et
Facile.* Cajori writes (vol. 1, pages 345-346):

In 1636 James Hume brought out an edition of the algebra of Vieta, in which he introduced a superior notation, writing down the base and elevating the exponent to a position above the regular line and a little to the right. The exponent was expressed in Roman numerals. Thus, he wroteIn 1637 exponents in the modern notation (although with positive integers only) were used by Rene Descartes (1596-1650) inAfor^{iii}A^{3}. Except for the use of Roman numerals, one has here our modern notation. Thus, this Scotsman, residing in Paris, had almost hit upon the exponential symbolism which has become universal through the writings of Descartes.

Descartes wrote: "*aa* ou *a*^{2} pour multiplier
à par soiméme; et *a*^{3} pour le multiplier
encore une fois par *a,* et ainsi à l'infini" (Cajori 1919,
page 178).

**Negative integers as exponents** were used by Nicolas Chuquet
(1445?-1500?) in 1484 in *Le Triparty en la Science des
Nombres.* Chuquet wrote to indicate 12*x*^{-1} (Cajori vol. 1, page 102).

Negative integers as exponents were first used with the modern notation by Isaac Newton in June 1676 in a letter to Henry Oldenburg, secretary of the Royal Society, in which he described his discovery of the general binomial theorem twelve years earlier (Cajori 1919, page 178).

Before Newton, John Wallis suggested the use of negative exponents but did not actually use them (Cajori vol. 1, page 216).

**Fractions as exponents.** The first use of fractional
exponents (although not with the modern notation) is by Nicole Oresme
(c. 1323-1382) in *Algorismus proportionum.* Oresme used to represent 9^{1/3}.
According to Cajori (1919), this notation remained unnoticed.

Simon Stevin (1548-1620) "had no occasion to use the fractional index notation," but "he clearly stated that 1/2 in a circle would mean square root and 3/2 in a circle would indicate the square root of the cube" (Boyer, page 356).

John Wallis (1616-1703), in his *Arithmetica infinitorum* which
was published in 1656, speaks of fractional "indices" but does not
actually write them (Cajori vol. 1, page 354).

Fractional exponents in the modern notation were first used by Isaac Newton in the 1676 letter referred to above (Cajori 1919, page 178).

**Scientific notation.** The earliest use of scientific notation
is not known. However, some physicists working with electricity in
in the decade or so up to 1873, when our modern volt, ohm, etc., were
standardized, used scientific notation. James A. Landau has found
only two usages of scientific notation in Maxwell's collected papers,
and could find no other physicists of mid-century using scientific
notation.

In 1863, in *The Annual Encyclopedia and Register of Important
Events of the Year 1862* the article on "Electricity" has on page
404:

The aim should be to make this standard [of electrical resistance] correspond to a current force equal to 10,000,000,000 times the value given by the quotient of 1 metre by 1 second of time, that is, 10In 1868^{10}mètre/seconds.

The above quotation was taken from the OED2.

In 1885 Johann Jakob Balmer in "Notiz uber die Spectrallinien des
Wasserstoffs" (*Annalen der Physik and Chemie,* Vol. 25, p. 80,
1885) wrote in English translation:

From the formula we obtained for a fifth hydrogen line49/45 · 3645.6 = 3969.65 · 10^{-7}mm

In 1887 Albert Abraham Michelson and Edward Williams Morley wrote in
*Philosophical Magazine,* Series 5, December, 1887:

Considering the motion of the earth in its orbit only, this displacement should beBoth of these citations were taken from2 D vThe distance D was about eleven metres, or 2 x 10^{2}/V^{2}= 2D x 10^{-8}^{7}wave-lengths of yellow light.

An even earlier possible use of scientific notation is by Robert
Whillhelm Bunsen in 1857 in *Philosophical Transactions,* where
these formulae appear on page 357:

I = IHowever the objection is that Bunsen was measuring the intensity of light before and after going through a tube of chlorine, and the alpha above is defined as "The value of 1/alpha, which signifies...the depth of chlorine to which the chemical rays must penetrate in order to be reduced to one-tenth of their original amount..." Therefore the 10 is not necessarily part of scientific notation but comes from the fact that Bunsen elected to measure a reduction of light to one-tenth of the original._{0}10^{-hα}I = I

_{1}10^{-hα}+ I_{2}10^{-hα}+ ...

This entry was largely contributed by James A. Landau.

In 1892 in *Mental Arithmetic,* M. A. Bailey advises avoiding
expressions containing both ÷ and ×.

In 1898 in *Text-Book of Algebra* by G. E. Fisher and I. J.
Schwatt, *a*÷*b*×*b* is interpreted as
(*a*÷*b*)×*b.*

In 1907 in *High School Algebra, Elementary Course* by Slaught
and Lennes, it is recommended that multiplications in any order be
performed first, then divisions as they occur from left to right.

In 1910 in *First Course of Algebra* by Hawkes, Luby, and
Touton, the authors write that ÷ and × should be taken
in the order in which they occur.

In 1912, *First Year Algebra* by Webster Wells and Walter W.
Hart has: "Indicated operations are to be performed in the following
order: first, all multiplications and divisions in their order from
left to right; then all additions and subtractions from left to
right."

In 1913, *Second Course in Algebra* by Webster Wells and Walter
W. Hart has: "*Order of operations.* In a sequence of the
fundamental operations on numbers, it is agreed that operations under
radical signs or within symbols of grouping shall be performed before
all others; that, otherwise, all multiplications and divisions shall
be performed first, proceeding from left to right, and afterwards all
additions and subtractions, proceeding again from left to right."

In 1917, "The Report of the Committee on the Teaching of Arithmetic
in Public Schools," *Mathematical Gazette* 8, p. 238,
recommended the use of brackets to avoid ambiguity in such cases.

In *A History of Mathematical Notations* (1928-1929) Florian
Cajori writes (vol. 1, page 274), "If an arithmetical or algebraical
term contains ÷ and ×, there is at present no agreement
as to which sign shall be used first."

Modern textbooks seem to agree that all multiplications and divisions should be performed in order from left to right.

**X for vector product** was
used in 1902 in J. W. Gibbs's *Vector Analysis* by E. B. Wilson.

**Plus-or-minus symbol (±)** was used by William Oughtred
(1574-1660) in *Clavis Mathematicae,* published in 1631 (Cajori
vol. 1, page 245).

**Postfix notation** or **RPN** began as prefix notation, a
mathematical notation which did away with grouping symbols. It was
proposed by Jan Lukasiewicz (1878-1956). "Prefix" meant that the
operators ( * + - etc.) preceded the operands or variables they were
meant to operate on. Then it was discovered that it is much more
convenient to place the operands first and operators last, so
"Postfix" or "reverse Lukasiewicz" or "reverse Polish" notation was
created. With postfix notation the operators themselves become
delimiters between operations. Thus the sequence U*(V^(W + 3))/(X -
Y) becomes

RPN is used in the computer language Forth. [Axel Harvey]

**The product symbol** () was introduced by Rene Descartes, according to Gullberg.

Cajori says this symbol was introduced by Gauss in 1812 (vol. 2, page 78).

**Square root.** The first use of was in 1220 by Leonardo of Pisa in
*Practica geometriae,* where the symbol meant "square root"
(Cajori vol. 1, page 90).

The radical symbol first appeared in 1525 in *Die Coss* by
Christoff Rudolff (1499-1545). He used (without the vinculum) for square roots. He did not use
indices to indicate higher roots, but instead modified the appearance
of the radical symbol for higher roots.

It is often suggested that the origin of the modern radical symbol is
that it is an altered letter *r,* the first letter in the word
*radix.* This is the opinion of Leonhard Euler in his
*Institutiones calculi differentialis* (1775). However, Florian
Cajori, author of *A History of Mathematical Notations,* argues
against this theory.

In 1637 Rene Descartes used , adding the
vinculum to the radical symbol *La Geometrie* (Cajori vol. 1,
page 375).

Placement of the index within the opening of the radical sign was
suggested in 1629 by Albert Girard (1595-1632) in *Invention
nouvelle.* He suggested this notation for the cube root (DSB;
Cajori vol. 1, page 371).

According to Cajori (vol. 1, page 372) the first person to adopt
Girard's suggestion and place the index within the opening of the
radical sign was Michel Rolle (1652-1719) in 1690 in *Traité
d' Algébre.*

However, a history note in a high school textbook states that the
symbol was first used by Girard "around 1633" (*UCSMP Advanced
Algebra,* 2nd ed., 1996, page 496).

According to Tropfke, *Geschichte der Elementarmathematik,* 4th edition, vol. 1: *Arithmetik und Algebra,*
Berlin/New York: De Gruyter, 1980, p. 296, Girard introduced this use with examples for the cubic and
the quintic root on fol. 1r° of the *Invention nouvelle*
and suggested on the same page also (3/2)49 for the fractional exponent of 49 [3/2 written as fraction].
[This information contributed by Siegmund Probst.]

Examples of Leibniz's use of the radical sign with index placed in the opening:

VII,3 N. 57_2 p. 736 (End of April 1676)

VII,3 N. 58 p. 752 (May 3, 1676)

VII,3 N. 60 p. 760 (June 26, 1676)

See http://www.nlb-hannover.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII3C.pdf.

In the *Mathematical Gazette* of Feb. 1895, G. Heppel wrote,
"Following Chrystal, Todhunter, Hall and Knight, and the majority of
writers [sqrt]*a* should be considered a quantity having one and
not two values, although the algebra of C. Smith and the article by
Professor Kelland in the *Encyclopedia Britannica* make
[sqrt]*a* have two values."

**Summation.** The summation symbol () was first used by Leonhard Euler
(1707-1783) in 1755:

Quemadmodum ad differentiam denotandam vsi sumus signo [capital delta], ita summam indicabimus signo ().The citation above is from

This symbol was used by Lagrange, but otherwise received little attention during the eighteenth century (Cajori vol. 2, pages 61 and 265.)

**Absolute value of a difference.** The tilde was introduced for
this purpose by William Oughtred (1574-1660) in the *Clavis
Mathematicae* (Key to Mathematics), composed about 1628 and
published in London in 1631, according to Smith, who shows a reversed
tilde (Smith 1958, page 394).

**Matrices.** In 1841, Arthur Cayley (1821-1895) used the modern
notation for the determinant of a matrix, a single vertical line on
both sides of the entries. The notation appeared in the *Cambridge
Mathematical Journal,* Vol. II (1841), p. 267-271. However,
Cayley used commas to separate entries within rows (Cajori vol. 2,
page 92).

The double vertical line notation was introduced by Cayley in 1843 (Cajori vol. 2, page 95).

In 1846, the first occurrence of both the single vertical line
notation for determinants and double vertical lines for matrices is
found in "Mémoire sur les hyperdéterminants" by Arthur
Cayley in *Crelle's Journal* (Cajori vol. 2, page 93).

Cajori (vol. 2, p. 103) writes that round parentheses were used for
matrices by many, including Maxine Bocher in 1919 in *Introduction
to Higher Algebra* and G. Kowalewski in 1909 in
*Determinantentheorie* (although Kowalewski also used double
vertical lines and a single brace).

Cajori also shows a use of brackets for matrices (and no commas
within) by C. E. Cullis in *Matrices and Determinoids* in 1913.

**Arrow notation.** In 1936 in *L'Agebre Abstraite* Oystein
Ore wrote:

Nous dirons que deux systemes algebriques S et S' sont homomorphes (par rapport a l'addition et a la multiplication) s'il existe une correspondance a —> a' entre les elements de S et S' donnant a chaque elementade S une image uniquea'dans S' telle que chaque element de S' soit l'image d'au moins un element de S et en outre telle que de a —> a', b --> b' on puisse conclure

As early as 1939 Bourbaki used the arrow in element-to-element notation ["la application x —> f(x)"].

Saunders Mac Lane wrote:

At first the vivid arrow notation f : X —> Y for a map was not available, and homomorphisms of homology groups (or rings) were always expressed in terms of the corresponding quotient group or rings. Thus the familiar long exact sequence of the homotopy groups of a fibration was originally described in terms of subgroups and quotient groups; this is the style used by all three discoveries of the sequence and of the covering homotopy theorem [...] The occurrence of exact sequences of homology groups (though not the name 'exact') was first noted by W. Hurewicz in 1941 [...] The practice of using an arrow to represent a map f : X —> Y arose at the same time. I have not been able to determine who first introduced this convenient notation; it may well have appeared first on the blackboard, perhaps in lectures by Hurewicz and it is used in the Hurewicz-Steenrod paper, submitted November 1940 ...The above quotation is from Saunders Mac Lane, "Concepts and Categories in Perspective,"

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